In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms. The derivation of a theorem is often interpreted as a proof of the truth of the resulting expression, but different deductive systems can yield other interpretations, depending on the meanings of the derivation rules.
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Iterated logarithm
In computer science, the iterated logarithm of n, written log* n (usually read "log star"), is the number of times the logarithm function must be iteratively applied before the result is less than or equal to 1. The simplest formal definition is the result of this recursive function: On the positive real numbers, the continuous super-logarithm is essentially equivalent: but on the negative real numbers, log-star is 0, whereas for positive x, so the two functions differ for negative arguments.
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Ramsey theory
This article provides an introduction. For a more detailed and technical article, see Ramsey's theorem. Ramsey theory, named after the British mathematician and philosopher Frank P. Ramsey, is a branch of mathematics that studies the conditions under which order must appear. Problems in Ramsey theory typically ask a question of the form: "how many elements of some structure must there be to guarantee that a particular property will hold?"
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Graph of a function
In mathematics, the graph of a function f is the collection of all ordered pairs (x, f). In particular, if x is a real number, graph means the graphical representation of this collection, in the form of a curve on a Cartesian plane, together with Cartesian axes, etc. Graphing on a Cartesian plane is sometimes referred to as curve sketching.
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Order type
In mathematics, especially in set theory, two ordered sets X,Y are said to have the same order type just when they are order isomorphic, that is, when there exists a bijection f: X ¿ Y such that both f and its inverse are monotone (order preserving). (In the special case when X is totally ordered, monotonicity of f implies monotonicity of its inverse. ) For example, the set of integers and the set of even integers have the same order type, because the mapping preserves the order.
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Subsequence
In mathematics, a subsequence is a sequence that can be derived from another sequence by deleting some elements without changing the order of the remaining elements. For example, the sequence is a subsequence of . Given two sequences X and Y, a sequence G is said to be a common subsequence of X and Y, if G is a subsequence of both X and Y.
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Tuple
In mathematics and computer science, a tuple is an ordered list of elements. In set theory, an (ordered) -tuple is a sequence (or ordered list) of elements, where is a positive integer. There is also one 0-tuple, an empty sequence. An -tuple is defined inductively using the construction of an ordered pair. Tuples are usually written by listing the elements within parentheses "" and separated by commas; for example, denotes a 5-tuple.
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Noether's theorem
Noether's (first) theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. The theorem was proved by German mathematician Emmy Noether in 1915 and published in 1918. The action of a physical system is the integral over time of a Lagrangian function (which may or may not be an integral over space of a Lagrangian density function), from which the system's behavior can be determined by the principle of least action.
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